### Archives

### Recent blog posts

- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Generalizations of Martin’s Axiom and the well-met condition January 11, 2015
- Many diamonds from just one January 6, 2015
- Happy new jewish year! September 24, 2014
- Square principles April 19, 2014
- Partitioning the club guessing January 22, 2014

### Keywords

Antichain square principles Rado's conjecture Partition Relations PFA(S)[S] Universal Sequences stationary hitting Whitehead Problem 11P99 ccc Generalized Clubs Singular Density Weakly compact cardinal Rainbow sets Fat stationary set diamond star Aronszajn tree Fast club Ostaszewski square Prikry-type forcing Coherent tree Almost Souslin Singular coﬁnality Martin's Axiom Almost countably chromatic xbox Club Guessing Forcing P-Ideal Dichotomy weak diamond Sakurai's Bell inequality OCA Knaster S-Space Forcing Axioms Singular cardinals combinatorics square Cardinal Invariants projective Boolean algebra polarized partition relation Uniformization Foundations HOD reflection principles Minimal Walks Shelah's Strong Hypothesis Parameterized proxy principle Slim tree sap Kurepa Hypothesis Singular Cofinality PFA weak square b-scale free Boolean algebra Diamond Successor of Regular Cardinal Stevo Todorcevic Mandelbrot set Jonsson cardinal Small forcing Poset tensor product graph Successor of Singular Cardinal 05A17 Non-saturation Cardinal function Absoluteness Dushnik-Miller very good scale Hedetniemi's conjecture Erdos-Hajnal graphs coloring number Axiom R 05D10 incompactness Souslin Tree Commutative cancellative semigroups L-space approachability ideal Hindman's Theorem Chang's conjecture Selective Ultrafilter Almost-disjoint famiy 20M14 Square-Brackets Partition Relations Hereditarily Lindelöf space middle diamond Chromatic number Rock n' Roll stationary reflection Constructible Universe Ascent Path Cohen real Prevalent singular cardinals Large Cardinals Reduced Power Microscopic Approach Fodor-type reflection Erdos Cardinal

# Author Archives: Assaf Rinot

## The current state of the Souslin problem

I gave a plenary talk at the 2017 ASL North American Meeting in Boise, March 2017. Talk Title: The current state of the Souslin problem. Abstract: Recall that the real line is that unique separable, dense linear ordering with no endpoints in … Continue reading

## 2017 Workshop in Set Theory, Oberwolfach

I gave an invited talk at the Set Theory workshop in Obwerwolfach, February 2017. Talk Title: Coloring vs. Chromatic. Abstract: In a joint work with Chris Lambie-Hanson, we study the interaction between compactness for the chromatic number (of graphs) and … Continue reading

Posted in Invited Talks
Tagged Chromatic number, coloring number, incompactness, stationary reflection
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## The eightfold way

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading

## Reflection on the coloring and chromatic numbers

Joint work with Chris Lambie-Hanson. Abstract. We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) … Continue reading

## Set Theory and its Applications in Topology, September 2016

I gave an invited talk at the Set Theory and its Applications in Topology meeting, Oaxaca, September 11-16, 2016. The talk was on the $\aleph_2$-Souslin problem. If you are interested in seeing the effect of a jet lag, the video is … Continue reading

## Strong failures of higher analogs of Hindman’s Theorem

Joint work with David J. Fernández Bretón. Abstract. We show that various analogs of Hindman’s Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that … Continue reading

## More notions of forcing add a Souslin tree

Joint work with Ari Meir Brodsky. Abstract. An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing … Continue reading

## Prikry forcing may add a Souslin tree

A celebrated theorem of Shelah states that adding a Cohen real introduces a Souslin tree. Are there any other examples of notions of forcing that add a $\kappa$-Souslin tree? and why is this of interest? My motivation comes from a … Continue reading

## The reflection principle $R_2$

A few years ago, in this paper, I introduced the following reflection principle: Definition. $R_2(\theta,\kappa)$ asserts that for every function $f:E^\theta_{<\kappa}\rightarrow\kappa$, there exists some $j<\kappa$ for which the following set is nonstationary: $$A_j:=\{\delta\in E^\theta_\kappa\mid f^{-1}[j]\cap\delta\text{ is nonstationary}\}.$$ I wrote there … Continue reading

Posted in Blog
Tagged reflection principles, square, stationary reflection, Weakly compact cardinal
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## Ordinal definable subsets of singular cardinals

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. A remarkable result by Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is a subset $x$ of $\kappa$ such … Continue reading

Posted in Preprints, Singular Cardinals Combinatorics
Tagged HOD, Singular cardinals combinatorics
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